Multiplier Weak-Type Inequalities for Maximal Operators and Singular Integrals

Abstract

We discuss a kind of weak type inequality for the Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operators that was first studied by Muckenhoupt and Wheeden and later by Sawyer. This formulation treats the weight for the image space as a multiplier, rather than a measure, leading to fundamentally different behavior; in particular, as shown by Muckenhoupt and Wheeden, the class of weights characterizing such inequalities is strictly larger than $A_p$. In this talk, I will discuss quantitative estimates obtained for $A_p$ weights, $p > 1$, that generalize those results obtained by Cruz-Uribe, Isralowitz, Moen, Pott and Rivera-Ríos for $p = 1$, both in the scalar and matrix weighted setting. I will also discuss an endpoint result for the Riesz potentials as well as recent work on the characterization of such weights.

Brandon Sweeting

NSF MPS-Ascend Postdoctoral Research Fellow, Mathematics

My current research interests are in harmonic analysis, specifically weighted norm inequalities for singular integral operators and Riesz potentials in both the scalar and matrix setting.